City of Hudson's weighed voting system under scrutiny

[jimrtex]

I'm quite sure the block split shown on the spreadsheet Torie posted is wrong.

The spreadsheet is clearly one intended for presentation, and since it was done in a way that shows part of Ward 1 and part of Ward 4 being shifted to Ward 2, someone might have carelessly assumed it was smaller parts.

The omission of the prison adjustment is troubling, but it might be on a following page.  Since the spreadsheet had 2000 populations, you might want the prison population to be included for comparability.

If Hudson did the block split correctly, the Hudson students might have assumed that it was the prison adjustment that got to the final numbers.

We don't know what populations were used by Lee Papayanopoulos to calculate the weights.   It appears that Columbia County administers elections according to the VTD boundaries.  There is no ambiguity with the 3rd Street boundary between the 2nd and 4th Ward.  All the residences to the west in Ward 2 will have addresses in the 200s.  And there is a clear physical separation between the the units in the Front Street block.  They may have a N or S address.

The boundaries between Wards 4 and 5, and Wards 3 and 5, may conform to actual practice, and should be used for determining weights, even if they don't precisely follow the charter.

[jimrtex]

The Common Council has 8 committees, each with 5 members.  With 40 committee slots, and 11 members, each member must serve on 3 or 4 committees, and 8 of 11 chair a committee.  One member actually serves on 5, but chairs none.  5 of the 8 chairs, only serve on two other committees.

Though the 8 committees each have 5 members, there are only two with representation from all 5 wards.  4 of the 8 committees have both aldermen from a ward on the committee.

One concern with weighted voting is than even if it works for voting in the full, body whether it denies representation to voters in committees.  Are the voters of Ward 5 denied committee influence that would be commensurate with their numbers?   Logically, they should have 2 members on every committee.

The 3 members who do not chair committees are from the smaller wards (1, 2, and 4).  Are they given a place at the table, but at the far end of the table?

Another concern with weighted voting is the amount of access voters have to their representative.  The voters in Ward 1 and 4 may see their alderman walking their dog every evening.   If we ignore the alleys, Ward 1 practically has about 6 blocks, as does Ward 4.  It may be harder to communicate with a alderman from Ward 5, both because of the distance, but also because of time constraints in sharing him with four times as many constituents.

If the council were reduced to 7 members (one alderman from 6 wards plus the president), the current committee structure could not be maintained.  Possibly 5 committees could, or some of the fluffier committees might be reduced to 3 members.

[jimrtex]

This shows the 2000 census populations with apparent block splits.  Note: ward boundaries are based on charter, rather than VTD, and apparent current election practice.



The green circle at the east end of Warren Street indicates an odd population in 2000.  The block in 2010 had a population of 0.

I can't really see a street that forms the southern boundary of the beak-like shape, and it appears that there is only a store such as convenience store at the intersection with some parking.  With a little more looking it is a garage, Todd Farrell's Car Care Center.

[jimrtex]

I will be expanding this post.

Weighted Voting Doesn't Work

'WEIGHTED VOTING DOESN'T WORK: A MATHEMATICAL ANALYSIS' by John F. Banzhaf IIl was published in the Rutgers Law Review in 1965.  This was just after the one man, one decisions, particularly Reynolds v Sims.  Many states previously had apportionment plans, particularly for their upper house, where counties were represented in a fashion analogous to the states in the US Senate.

The New Jersey legislature passed a plan providing each county with one senator, and a weighted vote ranging from 19.0 for Essex and 16.1 for Bergen, to 1.0 for both Sussex and Cape May.  The New Jersey Supreme Court struck down the plan as violating the New Jersey Constitution.   A district court in Washington state ordered a weighted plan for the legislature, but then the legislature enacted its own equal-population plan.  There were never any clear-cut court decisions suggesting that weighted voting violated equal protection.

In general, a voter's voting power is the inverse of the population-mumble-electorate of his district.  When districts are of equal population, each representative has equal voting strength and all voters have the same voting power.

If a district has a larger population, each voter in the district has a smaller voting power.  But in a weighted system this is cancelled by giving the representative a greater weight.  For example, a voter in a district of 50,000 has 1/5 the influence in the election of his representative, that a voter in in a district of 10,000.  But if the representative from the larger district is given five times the voting strength, the effect will be cancelled.

But Banzhaf concluded that this was not true, if the representatives voting strength is proportional to the population.

"In almost all cases weighted voting does not do the one thing which both its supporters and opponents assume that it does: weighted voting does not allocate voting power among legislators in proportion to the population each represents because voting power is not proportional to the number of votes a legislator may cast"

He provides some theoretical examples, which might be considered extreme.  In one case, there were five districts, one with 50,000 persons, and four with 10,000 each.  The representatives had a voting weights of A:5, B:1, C:1, D:1, and E:1, total:9.   Which ever way A voted would prevail since 5 is a majority of 9.  A was a dictator, and the other 4 were dummies since their vote did not matter.

Another example had four districts of 80,000, and one of 10,000, with voting weights of A:8, B:8, C:8, D:8, and E:1.  With 33 votes, 17 was a majority.  In this case, E had just as much power as the other four representatives, since any three would form a majority, regardless of their voting weight.

Banzhaf Power Index

The Banzhaf Power Index is calculated by considering all possible combinations of representatives.  For N representatives, there are 2^N combinations.

All combinations that represent a winning coalition (a majority of the total weighted vote) considered.

For A:5, B:1, C:1, D:1, E:1, total:9, majority:5, the winning coalitions are:

A alone: A
A in combination with any other: AB, AC, AD, and AE.
A in combination with any two others: ABC, ABD, ABE, ACD, ACE, and ADE.
A in combination with any three others: ABCD, ABCE, ABDE, and ACDE.
All five together: ABCDE.

Next the instances where a member is a critical are identified.  That is, if a member switching his vote causes the majority to be lost, he is a critical member.  In all 16 winning combinations, A switching his vote causes the majority to be lost.  No other members are ever critical.

The ratio of critical votes for a member, relative to the total number of critical votes is his voting power.  In this case, the voting power for A is 16/16 or 1, that is, A has absolute power.  The voting power for B, C, D, and E, is 0/16 or 0, that is, B, C, D, and E have no power.

Now consider the case of A:8, B:8, C:8, D:8, and E:1, total: 33, majority: 17.

The winning combinations are:

All combinations of 3: ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, and CDE.
All combinations of 4: ABCD, ABCE, ABDE, ACDE, and BCDE.
The combination of all 5: ABCDE.

No member is critical to four- and five-member combinations, as the remaining three members continue to form a majority.  But all members of the three-member coalitions are critical, as their removal causes the majority to be lost.

Thus A is critical to the five three-member combinations that it is part of, as are B, C, D, and E, for a total of 25 critical members.

Each member is critical 5/25 or 20% of the time, even though the voting weights for A, B, C, and D represent 24.2% of the total; and E only 3%.

[muon2]

In IL the political parties are organized based on weighted votes. Each elected committeeman (by precinct, or township in Cook, or ward in Chicago) has a weighted vote equal to the number of partisan ballots cast at the last primary. Those weighted votes are cast to elect the party chair of each county. The GOP also uses the weighted votes to elect the members of the state central committee, one per congressional district. On the state central committee the board members have weighted votes proportional to the number of partisan ballots cast in the primary.

An interesting issue with the system occurred last year when the GOP had to replace its state chair. The CDs had been educed from 19 to 18 for the 2012 primary, but the state central committee serves for 4 years and wasn't up until 2014. That caused the weighted vote to be tabulated for districts that no longer existed. As you can imagine there's a pretty large spread in GOP strength in the different districts. This was the weighting for the 19 CDs last year:

District 15 – 9.66
District 13 – 8.50
District 14 – 8.46
District 18 – 8.34
District 16 – 7.68
District 19 – 7.64
District 6 – 7.48
District 11 – 7.09
District 10 - 7.00
District 8 – 6.49
District 17 – 5.00
District 12 - 4.22
District 3 - 2.98
District 9 – 2.89
District 5 - 2.14
District 1 – 1.4
District 7 – 1.36
District 2 - 1.08
District 4  – .60

[jimrtex]

In IL the political parties are organized based on weighted votes. Each elected committeeman (by precinct, or township in Cook, or ward in Chicago) has a weighted vote equal to the number of partisan ballots cast at the last primary. Those weighted votes are cast to elect the party chair of each county. The GOP also uses the weighted votes to elect the members of the state central committee, one per congressional district. On the state central committee the board members have weighted votes proportional to the number of partisan ballots cast in the primary.

An interesting issue with the system occurred last year when the GOP had to replace its state chair. The CDs had been educed from 19 to 18 for the 2012 primary, but the state central committee serves for 4 years and wasn't up until 2014. That caused the weighted vote to be tabulated for districts that no longer existed. As you can imagine there's a pretty large spread in GOP strength in the different districts. This was the weighting for the 19 CDs last year:

District 15 – 9.66
District 13 – 8.50
District 14 – 8.46
District 18 – 8.34
District 16 – 7.68
District 19 – 7.64
District 6 – 7.48
District 11 – 7.09
District 10 - 7.00
District 8 – 6.49
District 17 – 5.00
District 12 - 4.22
District 3 - 2.98
District 9 – 2.89
District 5 - 2.14
District 1 – 1.4
District 7 – 1.36
District 2 - 1.08
District 4  – .60

Using the program lpdirect at Computer Algorithms for Voting Power Analysis I calculated the following:

IL- Congressional district.
Vote - Voting weight, which is presumably proportional to partisan vote total.
Rel.V - Relative voting weight (ie Voting weight divided by total voting weight of 10001)
Rel.P - Normalized Banzhaf Power Index
Dev. - Deviation (ie Rel.P/Rel.V - 1, expressed as percentage)

IL-  Vote  Rel.V   Rel.P  Dev.
15   966   0.097   0.099  2.29%
13   850   0.085   0.086  0.97%
14   846   0.085   0.085  0.92%
18   834   0.083   0.084  0.95%
16   768   0.077   0.077  0.22%
19   764   0.076   0.077  0.22%
 6   748   0.075   0.075  0.20%
11   709   0.071   0.071  0.24%
10   700   0.070   0.070  0.23%
 8   649   0.065   0.064  0.81%
17   500   0.050   0.049 -1.43%
12   422   0.042   0.041 -3.21%
 3   298   0.030   0.029 -1.97%
 9   289   0.029   0.028 -1.79%
 5   214   0.021   0.021 -2.33%
 1   140   0.014   0.014 -2.42%
 7   136   0.014   0.013 -2.34%
 2   108   0.011   0.011  1.40%
 4    60   0.006   0.006 -0.82%

With simple weighting, the voting power (as measured by the Banzhaf power index) of each member is roughly proportional to their voting weight, with a standard deviation of 1.49% and maximum deviation range of 5.50% it is well within accepted apportionment ranges.

I then "corrected" the weights by adjusting the simple weights by the inverse of the deviation.  That is, the weight for IL-5 was adjusted by 966 * (1 - 2.29%) => 944.

The deviation for all districts, but one, was reduced to under 1%.  IL-2 which originally had 1.40% too much voting power, was reduced to 106, and now had -4.91% too little power, increasing the maximum absolute deviation by over half, but reducing the standard deviation to 1.99%.

Iterating the process a second time, the results were much better,

IL-  Vote  Rel.V   Rel.P  Dev.
15   946   0.097   0.097  0.06%
13   842   0.085   0.085  0.02%
14   838   0.085   0.085  0.06%
18   827   0.083   0.083 -0.04%
16   766   0.077   0.077  0.02%
19   762   0.076   0.076  0.00%
 6   747   0.075   0.075 -0.02%
11   710   0.071   0.071  0.00%
10   701   0.070   0.070  0.02%
 8   653   0.065   0.065 -0.04%
17   507   0.050   0.050  0.03%
12   435   0.042   0.042 -0.08%
 3   302   0.030   0.030  0.08%
 9   293   0.029   0.029  0.18%
 5   218   0.021   0.021 -0.22%
 1   144   0.014   0.014 -0.06%
 7   139   0.014   0.014 -0.58%
 2   110   0.011   0.011 -0.08%
 4    61   0.006   0.006  0.18%

Note the deviation was measured relative to the original weight which was proportional to the partisan vote in the district.  The standard deviation was 0.16%.  IL-2 which had originally been told that 108 votes gave it too much power, ended up with 110.

[muon2]

I'm actually impressed that the IL GOP State Committee is as balanced as it is with the deviations between actual weight and Banzhaf power within 5%. My initial guess is that it is due to a suitably large group and a cluster of both high weight and low weight districts to share power. That leads me to speculate that perhaps the weighted vote becomes more efficient as the number of members rises.

[jimrtex]

I'm actually impressed that the IL GOP State Committee is as balanced as it is with the deviations between actual weight and Banzhaf power within 5%. My initial guess is that it is due to a suitably large group and a cluster of both high weight and low weight districts to share power. That leads me to speculate that perhaps the weighted vote becomes more efficient as the number of members rises.

I think your speculation is correct.

Consider, a five-member body, where each member represents a district of equal population.  For simplicity, we will used normalized voting weights, so each member has a voting weight of 20%.

A simple majority of the council actually requires a 3/5 super-majority.

Now let's start shifting voters from District A to District E, and adjusting the weights:

A..E: 19, 20, 20, 20, 21
A..E: 17, 20, 20, 20, 23
A..E: 14, 20, 20, 20, 26
A..E: 11, 20, 20, 20, 29

Even at this point, where Districts A and E have a population that deviates from the mean by 45% (or a 90% range), it still takes a 3-member coalition to create a majority vote.  If E is joined by two of B,C,D, the coalition has a 69% (more than 2/3) supermajority.  Even if E leaves and is replaced by A, the coalition still has a bare 51% majority.

To actually change the result, we would have to increase the weight of E, and/or reduce the weight of A, such that some 2-member coalitions prevail, and some 3-member coalitions fail.

But at least in this case, that would mean members who represent a minority of the population could prevail in a vote, or that members who represent a majority of the population might lose.

To produce the desired result, we might need to force some variation among B, C, and D, which is likely to occur naturally.

A..E: 12, 19, 20, 21, 28

We initially assign weights of 120, 190, 200, 210, and 280, which results in an equal power of 20% for all members.  The relative error for A is 66.7%.  I adjust by the multiplicative inverse to get a new weight of 120 / (1 + 66.7%) = 72.  Doing this for all 5 members and normalizing the weights so that they total to (approximately) 1000, I get the following weights.

A..E: 68, 170, 188, 207, 368

But those voting weights produce voting power of 0%, 16.7%, 16.7%, 16.7%, and 50%.

But if we instead correct for 10% of the error, that is:

120 / (1 + 66.7%/10) = 113

A..E 113, 189, 200, 211, 288

But these voting weights still produce equal power.  If I twiddle a bit to correct for 1/8 of the error, I get:

A..E 111, 189, 200, 211, 290

These produce voting power of 14.3%, 14.3%, 28.7%, 28.7%, 28.7%, which at least recognizes a difference among the extreme members, but also introduces a large discrepancy between B and C, even though they only differ in simple weight by 5%.  The voting power of 14.3% and 28.7% are 1/7 and 2/7.  This indicates that there are simply two few critical combinations to provide the resolution we need.

Going back to the weights of A..E: 11, 20, 20, 20, 29, there are 2⁵ or 33 combinations:

1 (0 members): 0% total weight, not winning.
1 (1): 11%, not winning.
3 (1): 20%, not winning.
1 (1): 29%, not winning.
3 (2): 31%, not winning.
4 (2): 40%, not winning.
3 (2): 49%, not winning.
3 (3): 51%, winning, 3 critical members in each combination.
4 (3): 60%, winning, 3 critical members in each combination.
3 (3): 69%, winning, 3 critical members in each combination.
1 (4): 71%, winning, no critical members.
3 (4): 80%, winning, no critical members.
1 (4): 89%, winning, no critical members.
1 (5): 100%, winning, no critical members.

There are only 30 critical members and each member is critical to 6 or 20%.  I would guess that when the weights did produce voting power of 1/7 1/7, 2/7, 2/7, and 2/7 there were 28 critical members.

The number of combinations are too few, too sparse, and too discontinuous to produce the resolution needed for effective weighting.

But if there are 19 members, there are over 1/2 a million combinations, with probably 1/2 winning.  We estimate the number of critical combinations as the number of 10-member combinations (19! / (10! * 9!) = 92,378.   This is a coarse approximation.  The 6 largest districts can form a majority, and conversely the 14 smallest are needed.  Since there are more 10-member combinations than 6 member or 14 member combinations, this represents an outer limit.

In some of these combinations, only the largest member will be critical.  That is, the remaining members had a total weight well short of 50%, and the largest member had enough weight to barely reach a majority.  The other uncommitted members could not produce a majority by themselves.  But the remaining members might have a total weight of almost 50%, and any uncommitted member would push them over the threshold.  So let's guess 5 critical members per winning combination or 92,378 * 5 = 461,890.

The program lpgenf at Computer Algorithms for Voting Power Analysis says the actual number is 798,584.

Because of the large number of combinations, winning combinations, and critical combinations, the total weight of the critical combinations likely approaches a continuous function.  In addition the number of combinations should reach a maximum near where the total weight of the combination is 50%.   If it is a maximum, the first derivative is zero, and the frequency in the region roughly constant.

The probability that a member of a critical combination will be a critical member then is proportional to its voting weight.

The continuity of the number of combinations for a given total weight will be less, if the weights are quantized.  In the case of the state committee there are some quantizing factors.  The congressional districts have (or had in 2000) equal population, and presumably similar electorates.  To the extent the districts were gerrymandered, they will have similar Republican support (ie a Democratic gerrymander will seek to crack Republicans into as many districts where they can't reach a majority, but eat up a lot of votes getting close even in Republican years).  There ware also the black districts where Republican support is in the teens.  IL-15 had more Republican votes than the lowest 6 districts combined.

The distribution on the committee is not so much a number of large districts (IL-15 is less than twice the mean), but the number of quite small districts.  While IL-12 might not elect a Republican congressman, it can certainly contribute to a gubernatorial or senatorial victory.

Because of the partisan electoral system in Hudson, with two aldermen elected per district, it may effectively function as a 5 or 6-member body.  The two alderman from each ward can not be considered to be independent (in the mathematical sense).

[jimrtex]

This is a counter-example.  The New Jersey legislature was to elect one senator from each of the 21 counties, with the senator's vote weighted in proportion to the county's population.  The smallest counties, Sussex and Cape May, were assigned a voting weight of 1.0, which I have multiplied by 10.  The largest county, Essex, had a voting weight of 19.0, which I converted to 190.  The total voting weight was 125.0, so it is possible that the legislature arbitrarily picked a whole number of votes, rather than starting from 1.0 for the smallest counties, or perhaps it was a combination of the two.

Swing is the number of instances that the county is critical to a winning combination - that is, removal of the count would cause the majority to lose.  A majority of 62.6 was used.  The total number of critical instances was 2,913,530

Iin 1965, Banzhaf used IBM 7094 at Columbia University to perform the calculations.  The computer was useful for other things as well Daisy Bell by H.A.L.

R.Pop is the relative population share - Essex County had 15.22% of the population.  Banzhaff used relative voting share, but I think population is the correct measure to use for computing errors.

R.Pow is the relative power (Swing / Total Swing) - Essex County had 477616 / 2913530 or 16.39% of the relative power. 

Dev. is the relative difference between power share and population share.  Essex County power share of 16.39% is 7.69% larger than its population share of 15.22%.

There appears to a bias towards the largest counties.  When the plan was originally passed, it was favored by smaller counties, who presumably believed they would retain a fair share of representation.

County           Vote Swing   R.Pop.  R.Pow    Dev. 
Essex            190  477616  15.22%  16.39%   7.69%
Bergen           161  386352  12.86%  13.26%   3.11%
Hudson           126  294382  10.07%  10.10%   0.37%
Union            104  239424   8.31%   8.22%  -1.13%
Middlesex         89  203540   7.15%   6.99%  -2.31%
Passaic           84  191704   6.70%   6.58%  -1.83%
Camden            81  184604   6.46%   6.34%  -1.95%
Monmouth          69  156466   5.51%   5.37%  -2.57%
Mercer            55  124274   4.39%   4.27%  -2.86%
Morris            54  121994   4.31%   4.19%  -2.90%
Burlington        46  103770   3.70%   3.56%  -3.75%
Atlantic          33   74230   2.65%   2.55%  -3.92%
Somerset          30   67460   2.37%   2.32%  -2.39%
Gloucester        28   63122   2.22%   2.17%  -2.52%
Cumberland        22   49452   1.76%   1.70%  -3.63%
Ocean             22   49452   1.78%   1.70%  -4.87%
Warren            13   29180   1.04%   1.00%  -3.89%
Salem             12   26942   0.97%   0.92%  -4.45%
Hunterdon         11   24654   0.89%   0.85%  -5.12%
Sussex            10   22456   0.81%   0.77%  -5.07%
Cape May          10   22456   0.80%   0.77%  -3.70%

I was curious whether the low resolution of the voting weights had much effect.  For the smallest counties, a rounding from 1.0499 to 1.0 is a 5% error.  I calculated new weights based on (approximately) 10,000 total votes, the same number as used for the Illinois state committee.  This provides one vote for each 1/100 of 1% of the population, which is 8 times finer than the resolution that had been proposed in the 1960's.

The standard deviation dropped from 2.90% to 2.77%.  Some of the smaller counties which had unfavorable rounding in the 1960's plan, such as Ocean, Salem, Hunterdon, Sussex, reduced their deviation by about 1%,  But the deviation for the largest county, Essex, increased because it had also had a downward rounding.

County          Vote  Swing   R.Pop.  R.Pow    Dev. 
Essex           1522  478620  15.22%  16.43%   7.91%
Bergen          1286  385656  12.86%  13.24%   2.92%
Hudson          1007  294200  10.07%  10.10%   0.31%
Union            831  239212   8.31%   8.21%  -1.22%
Middlesex        715  204372   7.15%   7.01%  -1.91%
Passaic          670  191216   6.70%   6.56%  -2.08%
Camden           646  184068   6.46%   6.32%  -2.23%
Monmouth         551  156216   5.51%   5.36%  -2.72%
Mercer           439  124088   4.39%   4.26%  -3.00%
Morris           431  121744   4.31%   4.18%  -3.10%
Burlington       370  104336   3.70%   3.58%  -3.22%
Atlantic         265   74548   2.65%   2.56%  -3.51%
Somerset         237   66508   2.37%   2.28%  -3.77%
Gloucester       222   62644   2.22%   2.15%  -3.26%
Cumberland       176   49480   1.76%   1.70%  -3.57%
Ocean            178   50056   1.78%   1.72%  -3.70%
Warren           104   29152   1.04%   1.00%  -3.98%
Salem             97   27212   0.97%   0.93%  -3.49%
Hunterdon         89   24896   0.89%   0.85%  -4.19%
Sussex            81   22736   0.81%   0.78%  -3.88%
Cape May          80   22508   0.80%   0.77%  -3.48%

I then adjusted the weights in an attempt to make the voting power (as measured by the Banzhaf Power Index, more proportional to the population).  I adjusted the population of each county, by multiplying the actual population by the multiplicative inverse of the deviation, and then calculated new voting weights.   Note that the deviation is measured relative to the actual population of the counties, and not the adjusted "population", which was only used to calculate the voting weights.

The standard deviation reduces from 2.77% to 0.48%.  It appears that there is an overcorrection, as the two greatest deviations, for Essex and Hunterdon, flipped signs.

County          Vote  Swing   R.Pop.  R.Pow    Dev. 
Essex           1409  446635  15.22%  15.01%  -1.38%
Bergen          1248  383843  12.86%  12.90%   0.32%
Hudson          1002  300119  10.07%  10.09%   0.21%
Union            840  247949   8.31%   8.33%   0.28%
Middlesex        728  213557   7.15%   7.18%   0.38%
Passaic          683  199845   6.70%   6.72%   0.23%
Camden           660  192839   6.46%   6.48%   0.31%
Monmouth         566  164523   5.51%   5.53%   0.33%
Mercer           452  130819   4.39%   4.40%   0.15%
Morris           444  128615   4.31%   4.32%   0.26%
Burlington       382  110439   3.70%   3.71%   0.32%
Atlantic         274   78851   2.65%   2.65%  -0.05%
Somerset         246   71031   2.37%   2.39%   0.66%
Gloucester       229   65753   2.22%   2.21%  -0.55%
Cumberland       182   52207   1.76%   1.75%  -0.36%
Ocean            185   53279   1.78%   1.79%   0.38%
Warren           108   31041   1.04%   1.04%   0.13%
Salem            100   28693   0.97%   0.96%  -0.34%
Hunterdon         93   26821   0.89%   0.90%   1.09%
Sussex            84   24165   0.81%   0.81%   0.05%
Cape May          83   23863   0.80%   0.80%   0.22%

Two more iterations, the last increasing the total vote weight to 100,000, produced a standard deviation of 0.13% (weighted by population 0.09%) and a range just under 0.5%.  Conceivably the adjusted population could be used as the voting weight, but that might cause extra concern or possible ridicule.  Could an understandable explanation of why Essex County with a population of 923,545 receives 866,090 be made?  Where did the 57,455 vote(r)s go?  Would it be perceived that it was because they were from Newark and black?  It may be simpler to give the representatives 14,257 votes and explain that it makes the voting power of the Essex senator proportional to its population.

County          Vote  Swing   R.Pop.  R.Pow    Dev. 
Essex          14257  452304  15.22%  15.23%   0.06%
Bergen         12462  382044  12.86%  12.87%   0.03%
Hudson         10000  298736  10.07%  10.06%  -0.07%
Union           8386  247000   8.31%   8.32%   0.07%
Middlesex       7255  212308   7.15%   7.15%  -0.02%
Passaic         6815  198856   6.70%   6.70%  -0.09%
Camden          6585  191740   6.46%   6.46%  -0.08%
Monmouth        5646  163760   5.51%   5.51%   0.05%
Mercer          4514  130372   4.39%   4.39%  -0.02%
Morris          4435  127988   4.31%   4.31%  -0.05%
Burlington      3816  110068   3.70%   3.71%   0.17%
Atlantic        2734   78640   2.65%   2.65%  -0.14%
Somerset        2446   70308   2.37%   2.37%  -0.19%
Gloucester      2301   66032   2.22%   2.22%   0.05%
Cumberland      1825   52392   1.76%   1.76%   0.17%
Ocean           1849   52956   1.78%   1.78%  -0.05%
Warren          1080   30864   1.04%   1.04%  -0.26%
Salem            999   28680   0.97%   0.97%  -0.20%
Hunterdon        927   26516   0.89%   0.89%   0.12%
Sussex           842   24132   0.81%   0.81%   0.10%
Cape May         829   23820   0.80%   0.80%   0.23%

[muon2]

I think you hit it on the head. A weighted vote that is based on a clear proportion of a quantity like population would seem fair to the public. The Banzhaf weight would lose most of the public, and deviations of a few percent in the power index wouldn't be of great concern. The fact that some representatives have slightly larger or smaller indices would be seen as a feature of the weighted voting system.

OTOH if a weighted vote was based on equal power indices and that caused votes to be noticeably different than the direct proportional share, the public would not be happy. There would be ridicule from some corners, and charges of a "fixed" system from others. If it was in place for a long time the charges might abate, but the media would perennially ask why the weighted vote didn't match the proportions.

[jimrtex]

I just came across this entry (from 2011), in the blog that started this thread.

Some History of the Weighted Vote

I haven't come across the specifics of how the weights are calculated in Hudson.  My assumption was that the President should have a voting strength based on 1/10 of the city's population, and each alderman 1/2 of the population of his ward.  This would give a total represented population of 1.1 times the actual population, with the president representing 0.1/1.1 or 1/11 of the total - the same as he would if there were equal population districts with no weighting.

In the 2000s plan, there was one instance where the two aldermen from a single ward had voting weights that differed by one.  Aldermen have different voting weights, depending on whether the vote is a simple majority; 2/3 supermajority; or 3/4 supermajority.  That the weight of a member varies slightly on the type of vote is likely to be viewed with incredulity.  That two members have different weights also would be treated skeptically.  In the 2000s, one alderman had one additional vote on a 2/3 vote (the alderman with the greater popular vote when elected, received the bonus vote).

There was an alternative to the current plan that would have applied a similar split, but would be used on simple majority votes.  There were some small adjustments to other voting weights.  These weights were said to produce results slightly more comparable to population. I think the votes opposed to the current plan were based on a preference for the alternative.

I then searched the blog for "weighted" and came across some interesting entries.

Forty Years of Weighted Votes

Note the pictures.  These demonstrate that a picture is worth a 1000 words

"where the "old Hudson" folks, who grew up there still control matters in Hudson, but perhaps for not much longer, holding the power and most of the very much sought after government jobs.

Or alternatively, that the Appalachians do extend through New York.

The State of the Weighted Vote

Eureka!

This entry has a link to Lee Papayanopoulos's study of voting weights.

He cites a number of court cases from around 1970 upholding weighted voting schemes in several New York counties.

I think that the blogger might not understand the term 'egality' which is the number of wards where the two alderman have the same weight.  An egality of 5, means the weights are identical for all 5 wards.

The Papanopoulos study has different ward populations than we have assumed:

Ward 1  770  v 828
Ward 2  1,281 v 1204
Ward 3  1,142 v 1142
Ward 4   725 v 744
Ward 5   2,485 v 2,485

He says: "Populations based on the decennial census, adjusted to exclude institutional inmates and to reconcile overlapping election districts."

These would require a change in the block splits:

Front Street block: Ward 1:2  131:234 v 73:292
Great Northern block: Ward 2:4  51:238 v 32:257

If we use Ward 2: 6 buildings x 8 units + 2 buildings x 12 units = 72 units; and Ward 1: 5 buildings x 8 units = 40.

365 x 72/(72 + 40) = 234.6

The block split was reversed.   Someone from Ward 2 should sue.

[jimrtex]

The State of the Weighted Vote

Eureka!

This entry has a link to Lee Papayanopoulos's study of voting weights.

He cites a number of court cases from around 1970 upholding weighted voting schemes in several New York counties.

I think that the blogger might not understand the term 'egality' which is the number of wards where the two alderman have the same weight.  An egality of 5, means the weights are identical for all 5 wards.

The Papanopoulos study has different ward populations than we have assumed:

Ward 1  770  v 828
Ward 2  1,281 v 1204
Ward 3  1,142 v 1142
Ward 4   725 v 744
Ward 5   2,485 v 2,485

He says: "Populations based on the decennial census, adjusted to exclude institutional inmates and to reconcile overlapping election districts."

These would require a change in the block splits:

Front Street block: Ward 1:2  131:234 v 73:92
Great Northern block: Ward 2:4  51:238 v 32:257

If we use Ward 2: 6 buildings x 8 units + 2 buildings x 12 units = 72 units; and Ward 1: 5 buildings x 8 units = 40.

365 x 72/(72 + 40) = 234.6

The block split was reversed.   Someone from Ward 2 should sue.

The populations used by the wards by Papayanopoulos, are the same that were used by Columbia County for the 5 Hudson wards, each of which also serves as a county supervisor district.

Columbia County first weighted in 1972, and has reweighted after every census.  Hudson first weighted in 1975, and did not update these until 2004.   Papayanopoulos did the calculation in 2004.   I would not be surprised if Papayanopoulos did the Columbia calculations as well.

The definition of VTD's is done by each state.  It is voluntary, and not all states have VTD's. Their use varies from state to state.  In Maine, for example, it appears that they have some relationship to legislative districts, at least within larger cities, such as Portland.

The census bureau is going to insist on the use of census blocks and would be quite insistent on not splitting census blocks to conform to ward boundaries.  They might be willing to add a visible feature such as a creek as a block boundary if it was used for a ward boundary.  But this would be prospective only.

Redistricting is going to be based on census blocks, if at all possible, which will tend over time to coerce precinct boundaries to conform to block boundaries.

The census bureau is going to want a single contact in a state.  Whoever that is in New York would coordinate with the NYSBOE, which in turn would coordinate with the county BOE. Columbia County Board of Elections appears to conduct elections for Hudson.  The Hudson election results are on the county site, and the ward map from the original blog post is from the county site.

The VTD's in Columbia County use a consistent nomenclature, including a number associate with each town/city.  The 3 VTD's for Hudson indicate that they based on the election precincts.

That is:
1-1, 2-1, and 4-1
3-1 and 3-2 and
5-1 and 5-2.

Ward 3 once upon a time had two precincts, which is plausible since it stretches from 3rd Street to the eastern edge of the city.

The VTDs which supposedly match the boundaries for Ward 3 and Ward 5, do not match the ward boundaries in the city charter.  In particular the boundary between Ward 3 and Ward 5 in the charter is along Columbia Turnpike. rather than Columbia Street to the eastern city limits; and the boundary between Ward 4 and Ward 5 is along 5th Street and 5th Street extended to the northern city limits, rather than taking a jog to the west at Prospect, to Short Street and then on out Harry Howard.

In particular:

VTD "Ward 3" includes the triangle between Columbia Street, Columbia Turnpike, and the eastern city limits from Charter Ward 5;

VTD "Ward 5" includes the block bounded by Prospect, 5th, Washington, and Short; the "block" bounded by Washington, 5th, "Clinton", and Short (this is not a census block, as Clinton does not actually continue west of 5th Street); and the residences on the south side of Harry Howard, north of Underhill Pond, but not including the new apartments further east, from Charter Ward 4.  There is also a house or two on the north side of Clinton at the northern end of 5th Street,

[jimrtex]

In 2002, the common council passed a resolution specifying the ward populations as:

Ward 1: 883
Ward 2: 1483
Ward 3: 1957
Ward 4: 829
Ward 5: 2372
Total: 7524

These 2000 numbers are also in the Hofstra report.

The 2000 VTD's were the same as the 2010 VTD's.  The VTD's correspond roughly to the wards.

VTD 1-2-4 3114
VTD 3 2005
VTD 5 2405

Ward 1

The population of the blocks wholly within Ward 1 is 746, from which we can infer a split of the Front Street block of Ward 1:Ward 2 = 137:225.   This ratio is quite similar to my estimated ratio of housing units of 5x8:(6x8 + 2x12) = 40:72.  In 2000, they got Front Street block correct.

Ward 2

The population of the blocks wholly within Ward 2 plus the 225 persons from the Front Street block was 1439.  We can get to the Ward 2 population by adding 44 persons from the Great Northern block.  This produces a split of Ward 2:Ward 4 = 44:251 which is similar to that for 2010.

There are a number of ways to allocate block populations between two areas (wards, towns, etc.)

(1) Proportional to area.  This works OK in residential areas with consistent lot sizes.  It is computationally the simplest, since calculating areas of polygons is quite trivial.

(2) Proportional to streets along edge of blocks.  IIRC, this was the method used by the Ohio legislature.  Ohio has a horrific mismatch between census blocks, and political boundaries.   This can also be automated.

(3) Proportional to housing units.  This is simple if you only have a few areas, such as in Hudson.  It may have small errors due to vacancies, differences in the number of bedrooms, etc. but likely more accurate than (1) or (2).

(4) Take an actual census.  Hard to do and expensive, and probably won't produce a much better result than (3).

I'd guess that the preferred method is (3).  The Common Council was concerned that no population was returned for the Firemen's Home, but this was likely an error.  A census block formed by the driveway to the home had zero population, which is correct.  The population of the Fireman's Home was included within the Great Northern block.  In 2000, the block population was 295, which included 212 persons in 101 housing units, 90 of which were occupied, and 11 vacant.  In addition, there were 83 persons in group quarters, which is presumably the Firemen's Home population, unless there is another nursing home.

It should be the 212 persons in housing units that would be allocated on the basis of housing units, rather than the 295 total population.  If Hudson did the latter, then they overestimated the Ward 2 population.   A better split might be 31:(181+83).

Ward 3

The VTD 3 population was 2005, and Ward 3 was 1957.  This discrepancy is of 48 matches the 48 persons living in the triangle between Columbia Street and Columbia Turnpike on the eastern edge of the city.  Columbia Turnpike is the ward boundary in the charter, and Columbia Street is the VTD boundary.

Ward 4

If we take the population of VTD 1-2-4 of 3114 and subtract the population of wards 1 and 2, 883 and 1483, respectively, we get 748, which is 81 less than that for Ward 4 (829).

The 81 persons are located west of 5th Street and 5th Street extended, within VTD 5.   There are 27 persons in a census block bounded by Prospect-5th-Washington-Short, and 54 persons to the immediate north and and south of Harry Howard.  This splits a block with 174 persons Ward 4:Ward 5 54:120,  There are residents along the southern edge of this block including Clinton, 6th, Glenwood, Oakwood, and around to Paddock, so the split is reasonable.  The apartments on Harry Howard were built between 2000 and 2010, and the block was split for the 2010 census, apparently using a walkway path connecting the school to the residential areas to the south.

Ward 5

If we take the VTD 5 population of 2405, add 48 persons from VTD 3, that are not in Ward 3, and subtract 81 persons in VTD 5, which are not in Ward 5, we reach the Ward 5 population of 2372.

I haven't found a direct link between the ward populations from the 2002 resolution and the voting weights established in 2004, but we can check whether they are consistent.

[jimrtex]

Hudson 2000 weights for simple majority 1011/2020 weighted votes:

Ward            Vote    Swing R.Pop.  R.Pow    Dev.  
Ward 1            94     126   5.87%   5.51%  -6.15%
Ward 1            94     126   5.87%   5.51%  -6.15%
Ward 2           184     222   9.86%   9.70%  -1.55%
Ward 2           184     222   9.86%   9.70%  -1.55%
Ward 3           266     304  13.01%  13.29%   2.17%
Ward 3           266     304  13.01%  13.29%   2.17%
Ward 4            94     126   5.51%   5.51%  -0.04%
Ward 4            94     126   5.51%   5.51%  -0.04%
Ward 5           278     366  15.76%  16.00%   1.48%
Ward 5           278     366  15.76%  16.00%   1.48%
President        188     258

This is based on the assumption that each alderman represents half of his ward's population, and is independent of the other alderman (the latter is a very bad assumption).  The president is not assumed to represent anyone, nor expected to have a particular voting power.  Instead his weight is adjusted so that the relative power of the alderman is about the same as their relative share of the population.  There are 2544 critical combinations, 258 of which the president is the critical vote, and 2288 total where an alderman is critical.  The relative power is based on this president-excluded number.  That is for a Ward 1 alderman, his relative power is 126/2288 = 5.51%, which is 6.15% less than the Ward 1 share of the population.

The Ward 1 population is 6.51% greater than that Ward 4, yet both have the same voting weight.  I suspect it is easier to have weights converge so that they produce the same number of critical combinations, than trying to have them separated to produce a variation in the number of critical combinations that matches their population.

If the weight for Ward 1 were increased to 95, the number of critical combinations would increase from 128 to 148, and the relative power would increase to 6.36%, which is 8.34% greater than the population share.  

If we used the same voting weights, but modified the populations of Wards 3, 4, and 5 to be based on the VTD boundaries, the deviation balloons, particularly for Ward 4, which would be almost 11% overpowered.  The standard deviation also increases from 2.96% to 5.57%.

So it appears that the 2000 voting weights were based on populations that were derived using the ward boundaries in the city charter.   But it also appears that the elections were conducted using boundaries between Wards 4 and 5, and 3 and 5 based on VTDs.  That is the corrrect population was used, but the wrong voters.

Hudson 2000 weights for 2/3 supermajority 1337*/2004 weighted votes:

Ward            Vote    Swing R.Pop.  R.Pow    Dev.  
Ward 1           106      83   5.87%   6.12%   4.31%
Ward 1           106      83   5.87%   6.12%   4.31%
Ward 2           175     127   9.86%   9.37%  -4.97%
Ward 2           175     127   9.86%   9.37%  -4.97%
Ward 3           249     179  13.01%  13.20%   1.50%
Ward 3           243     177  13.01%  13.05%   0.37%
Ward 4           105      75   5.51%   5.53%   0.40%
Ward 4           105      75   5.51%   5.53%   0.40%
Ward 5           282     223  15.76%  16.45%   4.33%
Ward 5           281     207  15.76%  15.27%  -3.16%
President        177     149

*The above is based on supermajority of 2/3 + 1 (1337), rather than 2/3 (1336).  Since the total waighted vote of 2004 is divisible by 3, a 2/3 supermajority is possible with no wasted fraction.  But using 1336 as the quota produces a larger error.  6 of the 10 alderman would have an  absolute deviation greater than 5% and the standard deviation increases from 3.42% to 5.18%.  Increasing the quota, reduces the total number of critical combinations from 1552 to 1505, but does not do so uniformly.  In fact, the number of critical combinations increases for some alderman.  

This would be explained if there were combinations that barely reached the quota, so that all members of that coalition were critical.  If the quota were increased, then that combination would no longer be a winning combination.  But an alderman that switched from the opposition to such a combination would be critical to the new combination.

I can't be sure that there was such a consideration calculating the voting weights.  The law that specifies the voting weights, provides the quorum for taking a 2/3 vote.  If there were not full attendance, then the voting power of the attending aldermen, and the relationship among them changes.

Another feature of the 2000 voting weights was that the aldermen from wards 3 and 5 were different (but only for a 2/3 vote).  The 6 vote difference between the two aldermen from Ward 3, produces only a small difference in their deviation (1.13%).  Meanwhile, a 1 vote difference for the two aldermen from Ward 5 produces a 7.49% difference in their deviation, as one becomes the most overpowered alderman, and the other the 3 most underpowered (the more powerful alderman was chosen on the basis of the popular vote, so at least there was some logic to the assignment).

Hudson 2000 weights for 3/4 supermajority 1555/2037 weighted votes:

Ward            Vote    Swing R.Pop.  R.Pow    Dev.  
Ward 1           103      43   5.87%   5.96%   1.50%
Ward 1           103      43   5.87%   5.96%   1.50%
Ward 2           199      73   9.86%  10.11%   2.59%
Ward 2           199      73   9.86%  10.11%   2.59%
Ward 3           220      93  13.01%  12.88%  -0.95%
Ward 3           220      93  13.01%  12.88%  -0.95%
Ward 4           100      39   5.51%   5.40%  -1.95%
Ward 4           100      39   5.51%   5.40%  -1.95%
Ward 5           318     113  15.76%  15.65%  -0.71%
Ward 5           318     113  15.76%  15.65%  -0.71%
President        193      57

The voting weights for a 3/4 supermajority produce the closest match between voting power and population, which is somewhat surprising given the fewer number of critical votes.  Perhaps the fact that two aldermen from Ward 5, or an alderman from Ward 3 in combination with one from Ward 5 can block a 3/4 supermajority has the effect of increasing the need to involve the aldermen from the smaller wards in producing a supermajority.

[jimrtex]

In Hudson, the voting weight of the president is permitted to vary so as make the voting power of the aldermen a better match the population of their districts.

In the 2000s, the president had a smaller voting weight than an Ward 2 alderman on majority and 2/3 votes, but less on 3/4 vote.  The president's power varied from 10.13% and 9.90% on majority and 2/3 votes, to a mere 7.32% on 3/4 votes.  In a conventional 10+1 city council, each member, has 9.09% power.  On a 3/4 vote, the president has 19.48% less power than he would have under a equal population system.

In calculating the voting power in Hudson, it is assumed that the two aldermen from each ward vote independently, and in fact would split their vote half the time. 

For the 2000s majority vote, each alderman in Ward 1 would be the critical vote in 126 coalitions.  But in 62 of those, he would have split from his fellow ward alderman.  For the other wards, the numbers are:

Ward 1: 62 of 126, 49.2%
Ward 2: 102 of 222, 46.0%
Ward 3: 129 of 304, 42.4%
Ward 4: 62 of 126, 49.2%
Ward 5: 174 of 320, 54.4%.

In the complementary situation, where the aldermen vote together, and one is critical - it is really the pair of them together that is critical.  Because the share of critical votes that involve split votes varies among the wards, this is a further distortion of the ward power.

If we assume that aldermen from the ward vote together, then there is a bad mismatch between their voting power and population.  The smallest wards are seriously underpowered.

Ward            Vote    Swing R.Pop.  R.Pow     Dev. 
Ward 1           188       4  11.74%   7.69% -34.45%
Ward 2           368      12  19.71%  23.08%  17.08%
Ward 3           532      16  26.01%  30.77%  18.30%
Ward 4           188       4  11.02%   7.69% -30.18%
Ward 5           556      16  31.53%  30.77%  -2.40%
President        188       4

Note the small number of critical votes when there are only 6 voting units.  This is directly related to the fact that there are only 2⁶ = 64 voting combinations.  It may not be practical to weight the votes of such a small number of voting units.

Because pairs of aldermen have the same weight, there are many voting coalitions that are similar.  We can swap alderman 1A for 1B in hundreds of coalitions and produce the same vote total, and same circumstances with respect to criticality.

If both aldermen from a ward have the same voting weight (usually, but not always true in Hudson), then there are only 3^{5 .} 2 = 486 possible vote totals.  In the 2000s, because wards 1 and 4 had the same weight, and the President had a weight twice as large as those two, there were only 3^{3 .} 7 = 189 possible vote totals.

In 2000, the voting weights were approximately integer multiples.  If we normalize to the smallest voting weights:

Ward 1: 1.00
Ward 4: 1.00
Ward 2: 1.96
President: 2.00
Ward 3: 2.83
Ward 5: 2.96

This produced a stair-step distribution of vote totals:



The large steps correspond to vote totals where the weights are roughly integer multiples:

4 x 1  +  3 x 2  +  4 x 3  +  1 = 23 steps from 0 to 2020.  Within each large step their are smaller steps associated with the small number of unique weights.

In the 2000s, there were 92 (of the total 2048) combinations that had a vote total of 1010, which was exactly half the total voting weight.  Any member, no matter how infinitesimal their voting weight, could then be critical if they switched to such a coalition.  But it may take a lot of manipulation to produce these sort of situations, and it quite likely leads to members who represent a majority of the population, not having a voting majority, and vice versa.

If the voting weights are proportional to population, the voting power does not match the population well.  Here are the results for the 2010 population, with one alderman per ward, and the President presumed to represent 1/10 of the population.

Ward            Vote    Swing R.Pop.  R.Pow     Dev. 
Ward 1           755       4  12.03%   8.33% -30.70%
Ward 2          1309       8  20.01%  16.67% -16.69%
Ward 3          1142       8  17.84%  16.67%  -6.55%
Ward 4           712       4  11.32%   8.33% -26.40%
Ward 5          2485      24  38.81%  50.00%  28.83%
President        640       4

The results are substantially better with more voting units.  For example, for the Columbia County (2010):

Town            Vote  Swing   R.Pop.  R.Pow     Dev. 
Hudson City 4    725  148236   1.15%   1.13%  -2.47%
Hudson City 1    770  157184   1.23%   1.19%  -2.63%
Hudson City 3   1142  232588   1.82%   1.77%  -2.85%
Hudson City 2   1281  260588   2.04%   1.98%  -2.97%
Taghkanic       1313  266652   2.09%   2.02%  -3.13%
Ancram          1574  320236   2.51%   2.43%  -2.96%
Austerlitz      1654  336740   2.63%   2.56%  -2.89%
Gallatin        1668  339636   2.65%   2.58%  -2.88%
Canaan          1710  347852   2.72%   2.64%  -2.97%
Hillsdale       1928  394296   3.07%   2.99%  -2.45%
Germantown      1955  399696   3.11%   3.03%  -2.48%
Clermont        1966  401872   3.13%   3.05%  -2.50%
Stuyvesant      2030  415080   3.23%   3.15%  -2.47%
New Lebanon     2305  472408   3.67%   3.59%  -2.24%
Hudson City 5   2485  508432   3.96%   3.86%  -2.41%
Stockport       2819  577976   4.49%   4.39%  -2.20%
Copake          3617  746344   5.76%   5.67%  -1.58%
Livingston      3646  752464   5.80%   5.71%  -1.56%
Chatham         4132  856128   6.58%   6.50%  -1.17%
Greenport       4176  865272   6.65%   6.57%  -1.17%
Ghent           5408 1137528   8.61%   8.64%   0.33%
Claverack       6024 1280900   9.59%   9.72%   1.42%
Kinderhook      8501 1954008  13.53%  14.83%   9.64%

The only large anomaly is Kinderhook, which has more than 3 times the population of the average town (or ward), and 40% more than the next largest town of Claverack.

[muon2]

This seems to bear out my conjecture that weighted voting improves with the number of units, and is weakened when there are large disparities between unit populations. The county results would argue in favor of a reduction of wards in Hudson.

This is one of the complexities of the Hudson situation. From a city council view having more wards but with even population and single representation would seem to be better. From the county view having fewer wards within their weighted system would be better. Ideally a single Hudson rep to the county makes sense since the population is smaller than Kinderhook with one rep. As an intermediate position, I wonder if NY law would permit 6 wards in the city but grouped in pairs resulting in three seats for Hudson in the county weighted vote.

[jimrtex]

Some other tidbits.

In 2011, the 5 Hudson supervisors, all Democrats, took over the Democratic caucus on the Columbia Board of Supervisors, braying "one man, one vote".  The four smallest wards are smaller than any town in the county, and even Ward 5 is only mid-range.

A pitfall with weighted systems is what happens in committees - with the composition of caucuses similar to committees.   Hudson has 8 5-member committees, with each alderman serving on 3 to 5 committees.  Most committees do not have a member from every ward - the most typical is representation from four wards, with double representation from one.   But ideally Ward 5 would have two members on every committee.   Its aldermen may have more voting power, but they don't have more time.

A couple of decades ago, the supervisor from Kinderhook resigned.  He was an independent, but ordinarily voted with the Democrats giving them an organizing majority.  It was anticipated that the Kinderhook council would name a Republican to fill the vacancy.  Meanwhile, the county charter prevented changing of committees in midyear, presenting the potential of committees passing measures that the body as whole would reject.   While this could happen in an equal vote body (eg Jumping Jim Jeffords), the voting power of the Kinderhook supervisor, roughly 1/8 of the total made the Columbia board more susceptible.

Supervisors in Columbia County are paid $14,000 per year, with towns augmenting that.   The 5 Hudson supervisors do not receive any extra, but there is the cost of 5 supervisors.

Not necessarily related to weighted voting, but interesting nonetheless: A group calling itself Vote Columbia, was urging part-time residents to register in Columbia County rather than New York City.   It is a two-hour train ride from Hudson to New York City, so commuting is possible, but adding time from home to/from the train stations, and allowing extra time to avoid missing hourly or less frequent trains, is going to add about 6 hours to a work day, so it is not convenient.

But it may be quite possible to commute on weekends, particularly if the job permits a late arrival on Monday and/or early departure on Friday.  Vote Columbia was advocating that people could keep a rent-controlled apartment in Manhattan, even if registered to vote upstate.

I seem to recall a similar effort to some town on the extreme northern edge of the Catskills, where it would unthinkable to commute, except on weekends, but those from New York City could vote on a proposal for a wind farm.

A Google search shows that "Vote Columbia" is now the website for the capital of South Carolina.

[jimrtex]

This is one of the complexities of the Hudson situation. From a city council view having more wards but with even population and single representation would seem to be better. From the county view having fewer wards within their weighted system would be better. Ideally a single Hudson rep to the county makes sense since the population is smaller than Kinderhook with one rep. As an intermediate position, I wonder if NY law would permit 6 wards in the city but grouped in pairs resulting in three seats for Hudson in the county weighted vote.

New York state appears to grant significant home rule authority to counties and cities, including the authority to devise its own form of government:

New York Municipal Home Rule Law section 10(1)(a)(13) begins with:

"(13) The apportionment of its legislative body and, only in connection with such action taken pursuant to this subparagraph, the composition and membership of such body, the terms of office of members thereof, the units of local government or other areas from which representatives are to be chosen and the voting powers of individual members of such legislative body."

There are some restrictions, in priority order:

a) "substantially equal weight for all the voters";
b) Only towns with more than 110% of a full ratio for a representative may be divided;
c) substantially fair and equal representation for political parties;
d) representative areas shall be convenient, contiguous, and as compact as practical.

Though most of these are oriented towards single member districts, they are not necessarily non-applicable to a weighted voting system as used in Columbia County.

An apportionment plan may provide that office holders of towns, etc. may also serve on a county legislative body.

The population base for apportionment may be residents, citizens, or registered voters (there may be court decisions that restrict use of the latter two, but the Supreme Court has never made a definitive decision, and ducked its last opportunity to make a ruling).

There is a possibility of a referendum, but that may be subject to a petition, or a provision of the charter, and parts of a plan placed before a referendum may be severable.

Only one plan per decade is permitted.  But it is possible to re-weight based on current valid data.   This would suggest that a plan based on votes cast could be valid.  There may be limitations on whether Hudson or Columbia County can restructure their governing bodies.  In Hudson, the voting weights are in the city charter, but the underlying principles are not.

Maybe the Hofstra students studied not only what changes might be made, but how they could be effected.  It is conceivable that some changes would require a lawsuit.

[jimrtex]

This seems to bear out my conjecture that weighted voting improves with the number of units, and is weakened when there are large disparities between unit populations. The county results would argue in favor of a reduction of wards in Hudson.

The underlying idea behind the Banzhaf Power Index is that all voting combinations are equally possible.   In effect, it is equivalent to each representative flipping a coin to decide their vote.

If there were 100 representatives each flipping a penny, and having a vote weight of 1% (and a total weight of 1 (or 100%), we'd expect over time that the the distribution of vote totals would form a normal distribution, with a maximum likelihood of 50 representatives voting for and 50 representatives voting against, to produce a total of 50%, just short of a majority.  Any member of the 50% who voted No could switch to Yes, and flip the result, and all members would be equally likely to have originally voted No, so that all members have equal voting power.

Now imagine if the representatives had a variable weight, from 2/3% to 1-1/3%.  We could keep the 100 representatives, or we could permit it to float a bit.   If there were more representatives with less than 1% voting weight, there could be more than 100 representatives.  In terms of apportionment, we could let districts grow until they reached more than 1-1/3% and then either split them, or adjust them by removing population.  If we let districts decline until they fall below 2/3%, we could merge them, or adjust them by adding population.

The distribution of total votes for each combination will still be close to a normal distribution, but we will also have vote totals between 49% and 50%, instead just at 49% or 50%.  The distribution will be more continuous.

A representative with a voting weight of w, will be able to flip the result if the total vote T is in the interval:
(50% - w,50%], and they were on the losing side.  For small values of w, the number of combinations in the interval is proportional to w, and the probability that the representative is on the losing side is 50%.   Voting power is equal to voting weight.

As w increases, the probability of the total vote being in the interval (50% - w,50%] does not increase as fast as w.  That is, as the vote total drops away from 50%, there are fewer and fewer combinations.  This would suggest that there might be a bias against larger voting weights.

But the probability that a representative who votes No, will be on the prevailing side (eg a motion is lost), increases with his voting weight.   Imagine a representative with a weight of 10%.  Since the votes are independent and random, we can treat him as voting first.

If he votes No, then the remaining representatives must vote 50%+ to 40%- for the motion to carry.  But half the time, the remaining representatives will produce 45% or fewer total votes.  We can think of the heavyweight as pushing a motion into being defeated, and then switching sides, and flipping the result.

If our 10% heavyweight votes No, then a not too implausible 40%:50% split by the remaining representatives would produce a 40:60 split.  But if the heavyweight switches, then he is critical to the success of the motion.  In the most likely split of the other representatives: 45% to 45%, he can flip the vote, along with perhaps a few middleweights.

Consider an even more extreme case, where a representative has a 30% voting weight.  If the Yes side has anything less than 30%, then this super-heavyweight necessarily voted No.  And yet he can save the day, even if the remaining representatives overwhelmingly vote against a motion by a 20%:70% margin.

There were 3 real-world examples of approximately 20 members.

In New Jersey, Essex County had 15.22% of the population, and 16.42% voting power; Bergen had 12.86% and 13.24%, and Hudson 10.07% and 10.10%.   The remaining 18 counties had less voting power than their voting weight.   It is likely that Hudson gained from the smaller counties, but lost to Essex.

In Columbia County, Kinderhook had 13.53% of the population, but 14.83% of the voting power (when using the population as the voting weight).  Claverack had 9.59% and 9.72%, and Ghent 8.61% and 8.64%.   The remaining 15 towns and 5 Hudson wards had less voting power than their share of the county population.  The relative error for Kinderhook (9.64%) is greater than that for Essex County (7.91%).   I suspect that is because Kinderhook is somewhat more of an outlier.  Kinderhook is 41% larger than Claverack, and 57% larger than Ghent; while Essex is 18% larger than Bergen and 51% larger than Union.  The ratio of Kinderhook to the median (Clermont) of 432% is also more than the ratio of Essex to the median (Burlington) of 410%.

On the Illinois Republican state committee, IL-15 had 9.66% voting weight and 9.88% voting power.  The next five districts had more voting power than their voting weight, but for all five,  the relative error was less than 1%, and even for IL-15 was only 2.29%.  IL-15 is only 149% more voting weight than the median IL-8.

This graph illustrates the distribution of the population for the units of the three bodies.  The vertical axis is the cumulative population (normalized to the total population), the horizontal axis is the units in rank order from smallest to largest, normalized to the number of units (0 percentile to 100 percentile).

New Jersey had the smoothest curve.  Columbia County is fairly linear for the smaller entities with a break towards the high end (it has a few exceptionally large towns), while Illinois Republican state committee is linear at the top end, with a break on the low end (it has a few districts that are exceptionally lacking in Republicans).

[jimrtex]

This is one of the complexities of the Hudson situation. From a city council view having more wards but with even population and single representation would seem to be better. From the county view having fewer wards within their weighted system would be better. Ideally a single Hudson rep to the county makes sense since the population is smaller than Kinderhook with one rep. As an intermediate position, I wonder if NY law would permit 6 wards in the city but grouped in pairs resulting in three seats for Hudson in the county weighted vote.

The size of Kinderhook is more problematic than the small size of the Hudson wards.

An alternative would be to split Kinderhook into 3 districts, pair the wards in Hudson: 2+4, 1+3, and 5, and split Claverack and Ghent into 2 districts each.

Or it might be possible to split Kinderhook into two parts based on Valatie and Kinderhook, and reduce the number of supervisors in Hudson to 3, which would reduce the board of supervisors by one.  Claverack is only 182% of the mean town size.

[jimrtex]

This is one of the complexities of the Hudson situation. From a city council view having more wards but with even population and single representation would seem to be better. From the county view having fewer wards within their weighted system would be better. Ideally a single Hudson rep to the county makes sense since the population is smaller than Kinderhook with one rep. As an intermediate position, I wonder if NY law would permit 6 wards in the city but grouped in pairs resulting in three seats for Hudson in the county weighted vote.

The size of Kinderhook is more problematic than the small size of the Hudson wards.

An alternative would be to split Kinderhook into 3 districts, pair the wards in Hudson: 2+4, 1+3, and 5, and split Claverack and Ghent into 2 districts each.

Or it might be possible to split Kinderhook into two parts based on Valatie and Kinderhook, and reduce the number of supervisors in Hudson to 3, which would reduce the board of supervisors by one.  Claverack is only 182% of the mean town size.

This is a quick version, splitting the Town of Kinderhook into Valaties and Van Buren based on VTDs.  Van Buren is based on the village of Kinderhook, and is named in honor of "Old Kinderhook" to avoid confusion with the town.  Hudson North was formed from Wards 2 and 4, Hudson South from Wards 1 and 3, and Hudson East from Ward 5.


Town            Vote  Swing   R.Pop.  R.Pow     Dev. 
Claverack       6024  707535   9.57%   9.95%   4.05%
Ghent           5408  624405   8.59%   8.78%   2.28%
Valatie         4861  558041   7.72%   7.85%   1.70%
Greenport       4176  473139   6.63%   6.66%   0.37%
Chatham         4132  468077   6.56%   6.58%   0.35%
Van Buren       3783  426345   6.01%   6.00%  -0.16%
Livingston      3646  410001   5.79%   5.77%  -0.38%
Copake          3617  406821   5.74%   5.72%  -0.36%
Stockport       2819  314119   4.48%   4.42%  -1.29%
Hudson East     2485  277323   3.95%   3.90%  -1.14%
New Lebanon     2305  257087   3.66%   3.62%  -1.20%
Stuyvesant      2030  225527   3.22%   3.17%  -1.58%
Hudson North    2006  222879   3.19%   3.14%  -1.57%
Clermont        1966  218431   3.12%   3.07%  -1.58%
Germantown      1955  217169   3.10%   3.06%  -1.59%
Hillsdale       1928  214297   3.06%   3.01%  -1.54%
Hudson South    1912  212609   3.04%   2.99%  -1.49%
Canaan          1710  188947   2.72%   2.66%  -2.12%
Gallatin        1668  184479   2.65%   2.60%  -2.02%
Austerlitz      1654  182949   2.63%   2.57%  -2.01%
Ancram          1574  173951   2.50%   2.45%  -2.10%
Taghkanic       1313  144395   2.09%   2.03%  -2.58%


The standard deviation is 1.60%, down from 2.63% in the original version.  The maximum deviation, for Claverack, is 4.05%, compared to 9.64% for Kinderhook in the original version.  The deviation for Claverack increased from 1.42% to 4.05%, which suggests that there is particular advantage for the very largest members.   A possible systematic rule would be to subdivide districts with an error of over 5%.

Had we applied this to Columbia County, initially keeping Kinderhook and Hudson whole, Kinderhook would have had a deviation of 8.05%, somewhat less than when Hudson was divided into 5 wards.  Splitting Kinderhook into Valatie and Van Buren leaves Hudson as the largest district, with an error of 3.82%, suggesting the possibility that it doesn't need to be split.

It appears, based on Kinderhook, that the supervisor, besides serving on the county board of supervisors, is the head of the 5-member town board.  The supervisor is elected every two years, while two council members are elected every two years, for a 4-year term, creating a 5-member board.

This might make the division into supervisor districts unworkable, or require separation of the position of supervisor separate from town president/chairman, etc.  Alternatively, the two top vote-getters for supervisor could serve on the county board, with the vote divided pro rata based on their popular vote.  The last two elections, the Kinderhook supervisor has run unopposed, but it is unlikely that would happen under a scheme of two being elected.

[muon2]

Earlier you cited state law that provides for sufficiently large towns to be split for county representation. But now it looks like a split town would create an issue of governance of the town board. I presume that state law also provides a resolution of this issue.

[jimrtex]

Earlier you cited state law that provides for sufficiently large towns to be split for county representation. But now it looks like a split town would create an issue of governance of the town board. I presume that state law also provides a resolution of this issue.

The default structure (ie before adoption of local home rule in 1963) in New York appears to be that towns have a supervisor and two or four councilmen  who make up the town board.  It may be possible to elect town councilmen by ward.  Also, towns with more than 50% of a county's population may have two supervisors, one who is the presiding supervisor.  It appears that this is mainly a provision to give such a town more weight on the county board of supervisors.

Cities have a mayor, a president of the town council, and a supervisor and alderman from each ward. 

The county board of supervisors is comprised of the supervisors of towns and cities of the counties.   A town supervisor serves a dual or even triple role (town presiding office, town legislator, and county legislator), while a city supervisor is only a representative on the county board.

New York appears to be a transitional area between New England and the areas to the west.  In New England the town is the local government, and counties have few or very limited powers.  They have been formally abolished in Connecticut..

The Old Northwest developed from south-to-north, so would have more influence from Virginians and Kentuckians who did not have towns, making the county the main local government, with towns existing within.  Also, towns in Old Northwest were not organic, instead being defined by the Public Land Survey System.  Towns in Michigan and Wisconsin have a larger role, at least in the administration of elections.  This may be due to their settlement from New England and New York, via the Erie Canal and Great Lakes.

The composition of county boards of supervisors would have been a gross violation of OMOV.  But the dual role of supervisors, would produce resistance to switching to a legislature model, with election from districts, and no formal link with the towns, or switching to a much smaller board of commissioners.  Areas with a stronger county government, would have malapportioned county commissioner districts, but those can simply be adjusted.

This would likely be the impetus for the development of the weighted voting system for counties in New York.  It preserves the traditional representation of towns (and cities) on county boards.

I haven't found the actual location on the Columbia County website, but if you web search for:
"Local Law" "weighted voting" "Columbia County", you should find a PDF file LocalLawNo.04-13.pdf that is the most recent Columbia County apportionment plan.  It is useful because it shows the basis on which the weights were measured.

This gives an explanation of how the current weights for Hudson were calculatedLee Papayanopoulos's study of voting weights.   Professor Papayanopoulos also did the calculation for Columbia County, but I have not come across any documentation of this.

The Columbia County local law states this:

"WHEREAS, the Columbia County Board of Supervisors wishes to amend Local Law No. 1 of 1972 thereafter amended by Local Law No. 1 of 1982 and Local Law No. 2 of 1993 and Local Law No. 2 of 2004 providing a plan of apportionment for the members of the Board of Supervisors of Columbia County on a weighted voting basis in accordance with the provisions of New York Municipal Home Rule Law section 10(1)(a)(13)."

New York State Code and Statutes.  Click on New York State Law, then MHR for Municipal Home Rule.  Section 10 is in Article II.  Section 10(1)(a)(1) to 10(1)(a)(12) are functional powers: incur debt, provide for roads and highways.   10(1)(a)(13) is the authority to apportion and organize the local government.  I'm pretty sure that it would give Columbia County a power to authorize a second supervisor for Kinderhook, as well as reorganizing representation for Hudson, with respect to the governing board.  Though it appears that this would be subject to mandatory referendum.

Note: general laws for counties, cities, and towns are found in statutes "CNT", "GCT", "SCC", and "TWN".  New York state only has 62 cities, the last of which, Rye, was granted its charter in 1942 (ie the state must charter a city, while a village may be formed by its inhabitants.

The provision you referred to appears to be intended to protect smaller towns from being split, and likely is more oriented towards counties that have adopted election of county boards from districts.  It is similar to provisions in constitutions intended to avoid splitting counties in the formation of legislative districts (eg Ohio or Texas),

[jimrtex]

My conjecture about the source of power appears to be correct.  The following is based on Monte Carlo methods (100,000 samples) and the 1960s proposed weighting for the New Jersey senate.

Pop.: Share of the state's population.

Range: Share of losing voting combinations within range of majority, based on the county population share.   For example, for Essex County, there are 35,007 losing combinations within 15.22% of 50% (34.78% to 50%).   For Cape May County, there are 2129 combinations within the range (49.20% to 50.0%).  For the state, there are 252,420 such losing combinations that are in range of a county (a losing combination with a total vote of 49.5% is in range for all 21 counties, would be counted 21 times in the 252,420 total.  The Range share is the county number of combinations to the state total.

If a losing vote result were displayed on a vote board, one could quickly determine which senators might be able to reverse the vote.  50% of combinations are losing, yet Essex County might be able to reverse 70% of the losing votes.   Nonetheless its range share is less than its population share, because there are fewer combinations around 35% total vote than 50%.

For the smaller and midsize counties, the range share and population share are quite comparable.  At first glance, it would appear that the larger counties have less opportunity to be swing voters.

Loser: For a senator to be a swing voter, the motion had to lose, it has to be in range of him being able to change the outcome unilaterally, and the senator had to have voted against the motion.   Since the random model assumes the senator will vote No on 1/2 the votes, it would be expected that 1/2 the time he voted No on a losing motion that he could flip.  But this is not true for heavyweights.  In effect, much of the time they caused the motion to fail.   If a motion receives as 40% Yes vote, it can be due to two reasons: (1) Essex voted Yes with its 15%, and the other counties voted 25%:60% against, which is quite unlikely; or Essex voted No with its 15%, and the other counties voted 40%:45% which is quite possible.

Of the 35,007 voting combinations that Essex County was in range to be able to flip, they voted No on 22,870, or 65.3% of them.   For smaller counties, the share was quite close to 50%.  Interestingly, Warren was a bit higher.  This is likely related to the discontinuity in the population distribution, with the five smallest counties, including Warren, having similar populations, and Ocean having considerably more.  There is a similar, though smaller gap between Atlantic and Burlington.

Power: The Banzhaf Power Index is the share of combinations for which a senator is a swing voter, relative to the total number of such voters, for the state.  While Essex was in range to be able to flip only 13.87% of such combinations, because it was on the losing side in 65.37% of them, it ended up being a swing voter in 16.49% of such combinations, vs its 15.22% population share.

Theor.: Power was calculated using Monte Carlo methods.  The theoretical value using all combinations was generated by the program lpgenf


County         Pop.  Range    Loser   Power   Theor.
Essex         15.22% 13.87%   65.33%  16.49%  16.39%
Bergen        12.86% 12.21%   59.44%  13.21%  13.26%
Hudson        10.07%  9.95%   55.76%  10.10%  10.10%
Union          8.31%  8.38%   53.98%   8.23%   8.22%
Middlesex      7.15%  7.29%   52.89%   7.02%   6.99%
Passaic        6.70%  6.88%   52.22%   6.54%   6.58%
Camden         6.46%  6.64%   52.02%   6.29%   6.34%
Monmouth       5.51%  5.71%   51.88%   5.39%   5.37%
Mercer         4.39%  4.61%   51.43%   4.31%   4.27%
Morris         4.31%  4.53%   50.61%   4.17%   4.19%
Burlington     3.70%  3.89%   50.43%   3.57%   3.56%
Atlantic       2.65%  2.78%   51.40%   2.60%   2.55%
Somerset       2.37%  2.47%   50.83%   2.29%   2.32%
Gloucester     2.22%  2.32%   50.62%   2.14%   2.17%
Cumberland     1.76%  1.86%   49.76%   1.68%   1.70%
Ocean          1.78%  1.88%   49.36%   1.69%   1.70%
Warren         1.04%  1.09%   52.22%   1.03%   1.00%
Salem          0.97%  1.01%   49.88%   0.92%   0.92%
Hunterdon      0.89%  0.93%   48.77%   0.83%   0.85%
Sussex         0.81%  0.85%   48.79%   0.76%   0.77%
Cape May       0.80%  0.84%   49.37%   0.76%   0.77%


This shows the distribution of the total votes cast for the various combinations.  The bin sizes is 2%, with the centra bin 49% to 51% of the maximum vote.  The vertical lines correspond to the center of the bin.  The dimple at the top is likely real, rather than random luck.  Votes from the largest counties will tend to push the distribution away from an expectation of an balanced vote.

The ripples on the side slopes might also be real.  The three largest counties have 38% of the total vote, and would be expected to vote together 1/4 of the time.  That makes for a very large thumb on the legislative scales.

[jimrtex]

Searching for: "local law" "weighted voting" "New York" "county" I came across some other counties:

Madison County

Its local law specifies the procedure for deriving the voting weights.  There is one supervisor for each town; and the City of Oneida elects 2 supervisors from at large from each of 2 supervisor districts.  The supervisor districts in Oneida are each comprised of three wards.  So there is certainly precedent for changing how Hudson is represented on the Columbia board.

In Madison, an initial simple apportionment is done based on one vote for every 49 persons or fraction thereof.  Since the total is exactly 1500, I suspect that the quota was derived from a preferred total.

The initial voting weights were then adjusted such that the effective voting power was proportional to the population share within 5%.  Separate calculations were done for 1/2, 3/5, 2/3, and 3/4 majority.  In all cases, the total is 1500.  In Columbia and Hudson, the total varies.
The Oneida supervisors from each district cast equal votes, so I suspect that their vote being even was another constraint.

The Town of Sullivan has about 20.9% of the population.  Its population-based share of the vote is 313/1500.  Its weight for the various majorities varies dramatically:

1/2: 280
3/5: 291
2/3: 347
3/4: 615! (astonishment, not factorial).

Ontario County

It simply provides the weights for 1/2 and 2/3 majority with no explanation.   The cities of Geneva and Canandaigua elect 3 and 2 supervisors, respectively, with each elected from a pair of wards.

Schenectady County

Schenectady County has a county legislature.  When the legislature was created in 1965, the county was divided into 4 districts, 2 in the city of Schenectady, and 2 comprised of towns outside the city.   The legislators are apportioned among the districts on the basis of population.  The nominal size of the legislature is 15, but is adjusted when the apportionment error for any district is greater than 7.5%.  In one decade, the council was reduced to 13 members (4, 3, 3, 3).

In the 2010 census, it was discovered that no size less than 24 would be sufficient to produce an error less than 7.5%.  Faced with a dilemma of greatly enlarging the legislature, or changing the districts, both of which would cause problems since the legislature is elected by halves every two years, the charter was changed to simply apportion 15 members and weight the votes.

District 1 (3) Weight 1.0572
District 2 (3) Weight 1.0799
District 3 (5) Weight 0.9939
District 4 (4) Weight 0.9048

As one would expect, with such a small number of members, and relatively little difference in the weights it did not matter.  It still takes 8 members to pass a motion.  The 7 weightiest members only have 49.6% of the vote, and the 8 lightest 50.4%.

I suspect that there would be a pretty good OMOV case, since the county itself asserted that a deviation of greater than 7.5% was too much, but essentially gives one vote to districts that are almost 10% under.

Cattaraugus County

Cattaraugus County has a county legislature with 21 members elected from 10 multi-member districts (3 x 1, 5 x 2, 1 x 3, 1 x 5).   The largest includes the city of Olean, and its surrounding town of Olean, plus a portion of a neighboring town.   The other districts were comprised of one or more towns and cities.   Several of the two-member districts are made up of a half dozen or so towns, and could easily be split into single-member districts if you weren't super fussy about the population.

At some time (2000?), weights were imposed on top of this, but they range from 87% to 1.13%.  I suspect that the larger two-member districts were because they couldn't get a perfect split.  But with weighting, that would not be a problem.

The population represented ranges from 4.13% to 5.37%, while the voting power ranges from 4.54% to 4.98%, with the smallest districts favored the most.

If there were no weighting, all members would have 1/21 of the total power of 4.76%.  Arguably this is a OMOV violation, as a 30% variation in population is recognized with a 9.6% variation in power.